Hello this is Professor Kitch.
Welcome to this webcast on sections 9.8 & 9.9 which cover the concept of effective stress.
When you finish this module you should be able to describe the concept of a soil skeleton
and how that relates to interparticle forces.
The effective stress equation is one of the most important equations in geotechnical engineering
and you should be able to write it from memory.
And you should be able to compute the vertical effective stress in both the case where the
water is static and the case where there is steady state flow.
We'll also revisit the concept of the coefficient of lateral earth pressure in this section
and when you've completed the section you should be able to not only define K, but also
use is to compute the horizontal effective stress.
You should be able to draw Mohr's circles for both the total and effective stresses
at a point.
And finally you should be able to compute the pore pressure not only below the ground
water table, but also within the capillary fringe.
As we know, soil is a particulate material made up of solid particles and void space.
When a stress is applied to the soil.
Part of that stress is transmitted through the solid particles and their particle to
particle contacts.
If the void space is filled with water, part of that stress is carried through pore water
pressure.
If we zoom into a place where two particles come into contact we can see that the inter-particle
stress is carried through particle to particle contact and the pore pressure acts to push
the two particles apart.
The term soil skeleton refers to the arrangement of solid particles in the soil and their interconnections.
Stresses are transferred from particle to particle through these inter-particle contacts.
In a saturated soil pore water fills all the voids.
The pore pressure is the same in all directions and acts normal to the surfaces of the solid
particles.
Therefore the pore pressure tends to reduce the interparticle stresses.
Let's take a close look at the contact area between to soil particles.
There will be some total normal stress, sigma-T, being transmitted at this contact boundary.
That stress consists of the weight of the material from the tributary area above this
contact point divided by the contact area, A.
The pore water pressure u, acts normal to particle surface in opposition to this total
stress.
The rest of the total stress is transferred across the particle boundary as interparticle
stress, sigma-c.
So Sigma-c is that stress carried by the soil skeleton through interparticle contact.
And the total stress, sigma-T is the sum of the interparticle stress, sigma-c and the
pore pressure, u.
Or if we can rewrite the equations as Sigma-c is equal to the total stress minus the pore
pressure.
This division of the total stress into the portion carried by pore pressure and the portion
carried by the soil skeleton through interparticle contacts is at the heart of the concept of
effective stress.
The effective stress is that portion carried by the soil skeleton, here represented by
the interparticle stress sigma-c.
As we'll see in a moment, the effective stress isn't exactly equal to sigma-c, but conceptually
they're the same.
Now the inter particle contacts transfer both shear an normal stress, so let's take a close
look at how the pore water pressure affects these two components of stress.
If sigma-T and tau are applied at the top of the interface there must be an equal and
opposite sigma-T and tau at the bottom if the interface.
we know that there's a pore pressure, u, acting normal to the contact and carrying part of
the total stress, thereby reducing the contact stress, sigma-c, which is carried across the
contact, sigma-c.
However, the shear stress acts normal to the contact surface and therefore normal to the
pore pressure.
Therefore the shear stress is not affected by the pore pressure and the full shear stress
is transferred across the interparticle contact.
So pore pressure affects normal stresses within the soil skeleton but not the shear stresses.
So now let's expand our discussion from contact stress to the concept of effective stress
and define the relationships among total stress, pore pressure and effective stress.
Let's consider a point A below the groundwater table where the soil is saturated.
As we just discussed, the total stress at point A is carried partly by the soil skeleton
through interparticle contact stresses.
And partly through the water by pore pressure.
We previously defined the interparticle stress, sigma-sub-c, based on the contact area between
particles.
While this definition was helpful for understanding how stresses are carried by the soil skeleton;
it's not a useful formulation to use in practice, because it's not possible to know the area
of all the interparticle contacts in a volume of soil.
Instead we are going to define the effective vertical stress, sigma-prime-z as the total
stress sigma-z minus the pore water pressure, u.
The effective stress, sigma-prime-z, is still that portion of the total stress that is carried
by the soil skeleton.
But instead of defining this stress based on the area of particle contacts (as we did
with sigma-prime-c) we define it using the entire cross-sectional area of the soil element.
We do this for two reasons: first it makes the effective stress easy to compute, and
second it works.
So this is the equation we will use to define the effective stress.
We will use the notation sigma-prime to denote effective stress and sigma to denote total
stress.
And the effective stress will always be the total stress minus the pore water pressure.
In this equation we've dropped the subscript z because the definition applies in any direction,
vertical horizontal or otherwise.
Note that the concept of effective stress applies only to normal stresses and not shear
stresses.
As we discussed earlier, the pore pressure always acts normal to shear stresses and therefore
doesn't affect them.
So when we're discussion total and effective stresses, we're always talking about normal
stresses.
So now let's compute the effective vertical stress at A in this example.
To do this we'll assume A is at a depth z below the ground surface, and a depth z-w
below the ground water table.
We'll also assume our soil has a unit weight of gamma-1 above the groundwater table and
gamma-2 below the ground water table.
If the groundwater is static and we install a piezometer at point A.
The water in the piezometer will rise to the ground water table.
From our previous module on geostatic stress we know that the total vertical normal stress
at point A is the integral of gamma dz from zero to z, or for uniform soil layers, the
summation of gamma-i H-i.
Then the total vertical stress at point A is gamma-1 times the quantity z minus z-w
plus gamma-2 times z-w.
The pore pressure at A will be simply gamma-water times the height water rises in the piezometer,
z-w.
And the effective vertical stress at A, sigma-prime-z, will be the total vertical stress, sigma-z
minus the pore pressure, u.
And for this example, that will be (gamma-1 times the quantity z minus z-w plus gamma-2
times z-w) (minus gamma-water times z-w).
Or if we want we can combine the last to terms and write that gamma-prime-z is equal to (gamma-1
times the quantity z minus z-w) minus (the quantity gamma-2 minus gamma-water times z-w).
This equation stating that the effective stress is equal to the total stress minus pore water
pressure in perhaps the single most important concept in all of geotechnical engineering
and one you should commit to memory.
We learned in an earlier section that we generally computed the horizontal stress using the coefficient
of lateral earth pressure.
Now that we've developed the concept of effective stress, we need to revisit that concept of
lateral earth pressure and treat it more thoroughly.
We stated earlier that the coefficient of lateral earth pressure was the ratio of the
horizontal normal stress to the vertical normal stress.
Now we need to be more specific.
We will define the coefficient of lateral earth pressure, K, as the ratio of the effective
horizontal stress to the effective vertical stress or sigma-prime-x over sigma-prime-z.
Or we can write that sigma-prime-x is equal to K times sigma-prime-z.
So our procedure to compute the horizontal effective stress is to.
Compute the vertical effective stress, sigma-prime-z, as sigma-z minus the pore pressure, u.
Then determine K from load history and geologic effects (covered more completely in chapter
17).
Then compute the effective horizontal stress, sigma-prime-x as K times sigma-prime-z.
The total horizontal stress, sigma-x, is then computed as sigma-prime-x plus u.
Notice that we cannot compute the total horizontal stress directly from the total vertical stress.
We must first compute the effective stresses and then add the pore pressure back in to
get the total stresses.
This is because we defined the coefficient of lateral earth pressure in terms of effective
stresses rather than total stresses.
An interesting and sometime counter intuitive thing happens above the ground water table.
So let's take a minute or two to look at this situation.
Below the groundwater table of phreatic surface, the soil is saturated and the pore pressure
is equal to gamma-water time z-w.
Z-w is positive, and u is positive.
The effective vertical stress is equal the total vertical stress minus the pore pressure,
u.
And since u is positive the effective vertical stress will always be less than the total
vertical stress.
Now as we learned in Chapter 7, above the phreatic surface there is a zone of capillary
rise where the soil remains saturated.
In this zone the pore pressure, u, is still equal to gamma-water times z-w.
However this zone is above the free water surface and z-w is negative, therefore the
pore pressure is also negative.
The fundamental equation for effective stress still applies; the effective stress is equal
to the total stress minus the pore pressure.
. However, in this zone the pore pressure is
negative and we'll be subtracting a negative value from the total stress sigma-z.
Therefore within the zone of capillary rise the effective stress will actually be greater
that the totals stress.
This is due to the negative pore water pressure in this zone.
Above the zone of capillary rise the soil becomes unsaturated.
In the zone the pore water pressure is no longer equal to z-w time gamma-water.
We have to use unsaturated soil mechanics in this zone to describe the soil behavior,
which is a topic beyond the scope of this course.
Let's quickly look at how Mohr circles compare between total and effective stress.
Assume we have determined this to be the Mohr circle for some point in the ground based
on total stress, and let's assume that point is below the ground water table with a positive
pore pressure.
recalling our equation for effective stress, sigma-prime equals sigma minus u, we know
that the pore pressure, u, acts the same in all directions, so all normal stress will
be reduce by the same value of u when computing the effective normal stress, regardless of
the orientation of the normal stress.
We also know that the pore pressure does not affect shear stresses.
That means that the positive pore pressure will simple have the effect of shifting the
entire Mohr circle to the left without changing the size of the circle.
And the magnitude of that shift will be equal to the pore pressure, u.
now if we're in a situation where the pore pressure is negative (such as in the capillary
zone above the groundwater table, then the effective stresses will be greater than the
total stresses and the effective stress Mohr circle will be shifted to the right by the
amount of the negative pore pressure, u.
For our final topic in this section, let's compute effective stresses for cases where
we have steady state flow, such as under the concrete dam shown here.
Now Section 9.9 of your text describes the concept of seepage forces and how they affect
total and effective stress.
While this concept is correct and a valid way to compute effective stresses under steady
state flow, I find it confusing and I think there's a much simpler way to compute effective
stresses under steady state flow.
So let's look at this flow net which you analyzed in chapter 8.
And we'll look just at Point A, though this method would work for any location within
this flow net.
The depth to point A is z.
And the depth of water ponded behind the dam is H-w.
Now we've installed a piezometer at point A and we know from flow net theory that the
height the water rises in that piezometer, z-w, is less than the total height to the
tailwater, z plus H-w.
That's because water is flowing from the upstream to downstream side of the dam.
so now that we have a piezometer installed at point A, all we have to do is use our equation
for effective stress, sigma-prime is equal to sigma minus u, to compute the effective
stress at A. First we compute the total vertical stress
at A.
The total vertical stress is the summation of Gamma times H which, in this case, is H-w
time gamma-water plus gamma time z.
The pore pressure at A is the height the water is rising in the piezometer at that point,
z-w times gamma-water.
This is exactly the same we compute pore water pressure for the static conditions.
The effective vertical stress is simply sigma-z minus u.
Or (H-w times gamma-water) (plus gamma times z) (plus z-w times gamma-water).
Notice that we didn't even discuss the topic of seepage forces.
This procedure is really no different than that used to compute effective stress under
static ground water conditions.
So this is Kitch's rule for computing effective stresses at any point under the ground.
Stick a piezometer at the location you're interested in and figure out how high the
water rises in that piezometer.
The pore pressure will be equal to the gamma-water times the height or the water in the piezometer.
Compute the totals vertical stresses as the summation of gamma time z.
Then compute the effective stress as the total stress minus the pore water pressure.
Keep it simple and stick to the fundamentals.
So let's review.
Total stress is divided into the effective stress which is that portion carried by the
soil skeleton.
And pore pressure carried by the pore water.
The fundamental equation is: the effective stress is equal to the total stress minus
the pore water pressure.
The coefficient of later earth pressure is the ratio of the effective horizontal stress
to the effective vertical stress and we compute the effective horizontal stress as the K time
the effective vertical stress.
Pore pressure affects normal stresses but not shear stresses.
In the saturated capillary zone above the ground water table the pore pressure is negative.
All the fundamental equations apply, just make sure you properly compute the pore pressure.
Use Kitch's rule to compute effective stress under steady state flow.
Stick a piezometer in.
Compute the pore pressure from the height of water in the piezometer.
Then compute the effective stress as sigma minus u.
Finally here's a deceivingly simple practice problem for you.
Compete it before class.
That's it!